Richard Courant > Quotes

“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.”
― Richard Courant, What Is Mathematics? An Elementary Approach to Ideas and Methods

“It becomes the urgent duty of mathematicians, therefore, to meditate about the essence of mathematics, its motivations and goals and the ideas that must bind divergent interests together.”
― Richard Courant

“For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: What is mathematics?”
― Richard Courant, What Is Mathematics? An Elementary Approach to Ideas and Methods

“Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality.”
― Richard Courant, What Is Mathematics? An Elementary Approach to Ideas and Methods

“mathematical objects states “only the relationships between mathematically ’undefined objects’ and the rules governing operations with them.” It doesn’t matter what mathematical things are: it’s what they do that counts. Thus mathematics hovers uneasily between the real and the not-real; its meaning does not reside in formal abstractions, but neither is it tangible. This may cause problems for philosophers who like tidy categories, but it is the great strength of mathematics—what”
― Richard Courant, What Is Mathematics?: An Elementary Approach to Ideas and Methods